Ok...so more precisely my question is : what is [tex] \lim_{x\rightarrow\infty} \frac{Li(x)}{x} [/tex] ? (I suppose this should give the same as [tex] \lim_{n\rightarrow\infty}\frac{S(n)}{n}[/tex] ?)
My hope would be that this limit is not 0...but I think it is.

You might expect the x/log(x) term to be the 'main' term of Li(x), so you should consider the limit

[tex] \lim_{x\rightarrow\infty} \frac{Li(x)}{x/\log(x)} [/tex]

You can use l'hopital to find that limit if you like or make use of the more satisfying inequality (both using Li(x) after you've integrated by parts once-or more if you want to adapt this to a more accurate asymptotic):